QUASICONFORMAL DEFORMATION OF THE SIERPINSKI GASKET

The figure shows a quasiconformal deformation of the classical Sierpinski gasket SG with smaller Hausdorff dimension. In [1] we proved that the infimum of the Hausdorff dimensions of all quasiconformal deformations of SG is equal to one. In fact, one can find a one-parameter family of planar self-similar iterated function systems whose invariant sets are all quasiconformally equivalent with SG and have Hausdorff dimension decreasing continuously from log 3/log 2 = dim SG towards one. See the following sequence of images.

Similar constructions can be made for other self-similar fractals of "gasket type", e.g., higher-dimensional Sierpinski-type gaskets and polygaskets.

[1] J. T. Tyson and J.-M. Wu, "Quasiconformal dimension of self-similar sets", Rev. Mat. Iberoamericana 22 (2006), 205-258.